Classification of Lie algebras with naturally graded quasi-filiform nilradicals

On extensions of free nilpotent Lie algebras of type 2

“…The multiplication table for g 6, 4 5,3 has a misprint, [Y, X 4 ] = 3X 4 must be declared instead of 4X 4 (otherwise, ad Y is not a derivation, in other words, Ja , c] = 0 fails for the terna (a, b, c) = (Y, X 1 , X 2 )); by doing the correctio, g 6, 4 5,3 is just r 1 2,3 . Proposition 3 provides the 2-sovable extension g 7, 1 5,3 which is equal to our r 4 2,3 if we consider the new basis {Y + Z, Y − Z; X 0 , X 1 , X 2 , X 3 , X 4 }.…”

“…The first algebra and the third one are just r 1,α 2,3 and r 2 2,3 . The second algebra is equal to r 3 2,3 by rescaling the basis of g 6,2 5,3 given [1] in the folowing way: {−Y ; X 0 , X 1 , X 2 , X 3 , −X 4 }. The multiplication table for g 6, 4 5,3 has a misprint, [Y, X 4 ] = 3X 4 must be declared instead of 4X 4 (otherwise, ad Y is not a derivation, in other words, Ja , c] = 0 fails for the terna (a, b, c) = (Y, X 1 , X 2 )); by doing the correctio, g 6, 4 5,3 is just r 1 2,3 .…”

“…5,3 given [1] in the folowing way: {−Y ; X 0 , X 1 , X 2 , X 3 , −X 4 }. The multiplication table for g 6,4 5,3 has a misprint, [Y, X 4 ] = 3X 4 must be declared instead of 4X 4 (otherwise, ad Y is not a derivation, in other words, Jacobi identity J 5,3 is just r 1 2,3 .…”

“…Section 4 is devoted to non-solvable Lie algebras with nilradical n 2,t ; this type of extensions have dimension at most one and are determined by derivations that centralize the Levi subalgebra of Der n 2,t . The results in Sections 3 and 4 will be used in the final section 5 to review and present in a unify way the classifications of Lie algebras with nilradical n 2,2 and n 2,3 given in [16] (only solvable extensions are considered) and [1]. The Lie algebra n 2,2 is the 3-dimensional Heisenberg algebra and n 2,3 is a 5-dimensional quasi-classical Lie algebra denoted in [1] as L 5,3 .…”

“…The results in Sections 3 and 4 will be used in the final section 5 to review and present in a unify way the classifications of Lie algebras with nilradical n 2,2 and n 2,3 given in [16] (only solvable extensions are considered) and [1]. The Lie algebra n 2,2 is the 3-dimensional Heisenberg algebra and n 2,3 is a 5-dimensional quasi-classical Lie algebra denoted in [1] as L 5,3 . Here quasi-classical means endowed with a symmetric non-degenerate invariant form according to [12,Definition 2.1].…”